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https://github.com/smallpondtom/uniquekronecker.jl
Unique Kronecker product operation implemented in Julia!
https://github.com/smallpondtom/uniquekronecker.jl
julia math
Last synced: about 2 months ago
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Unique Kronecker product operation implemented in Julia!
- Host: GitHub
- URL: https://github.com/smallpondtom/uniquekronecker.jl
- Owner: smallpondtom
- License: mit
- Created: 2024-09-11T06:11:02.000Z (3 months ago)
- Default Branch: main
- Last Pushed: 2024-10-16T12:51:01.000Z (2 months ago)
- Last Synced: 2024-10-18T08:19:52.720Z (2 months ago)
- Topics: julia, math
- Language: Julia
- Homepage: https://smallpondtom.github.io/UniqueKronecker.jl/
- Size: 1.88 MB
- Stars: 3
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
# UniqueKronecker.jl
[](https://sites.google.com/view/elizabeth-qian/research/ace-group)
[](https://github.com/smallpondtom/UniqueKronecker.jl/actions/workflows/CI.yml)
[](https://codecov.io/gh/smallpondtom/UniqueKronecker.jl)
[](https://smallpondtom.github.io/UniqueKronecker.jl/stable)
[](https://smallpondtom.github.io/UniqueKronecker.jl/dev)
[](LICENSE)
## Overview
**UniqueKronecker.jl** is a Julia package for computing non-redundant (unique) Kronecker products of vectors, generalizing to _n_ dimensions and _k_-repeated products. It provides utility functions to work with the associated coefficient matrices, enabling conversions between unique Kronecker products and their standard (possibly redundant) Kronecker counterparts.
### What is the Unique Kronecker Product?
The standard Kronecker product of a vector $\mathsf{\mathbf{x}} \in \mathbb{R}^n$ with itself, $\text{kron}(\mathbf{x}, \mathbf{x}) = \mathbf{x} \otimes \mathbf{x}$, produces all possible pairwise products of its elements, resulting in redundant terms when $x_i x_j = x_j x_i$.
The **unique Kronecker product**, denoted here as $\text{uniquekron}(\mathbf{x},\mathbf{x}) = \mathbf{x} \oslash \mathbf{x}$, eliminates these redundancies by considering only unique combinations of indices. For example:
For $\mathbf{x} \in \mathbb{R}^2$:
- **Standard Kronecker product**:
$$
\mathbf{x} \otimes \mathbf{x} = \begin{bmatrix} x_1^2 \\ x_1 x_2 \\ x_2 x_1 \\ x_2^2 \end{bmatrix}
$$
- **Unique Kronecker product**:
$$
\mathbf{x} \oslash \mathbf{x} = \begin{bmatrix} x_1^2 \\ x_1 x_2 \\ x_2^2 \end{bmatrix}
$$
Here, $x_1 x_2$ and $x_2 x_1$ are considered the same and included only once.
### Coefficient Matrices
The package provides functions to compute the associated coefficient matrices. For example, in a second-order Kronecker product (or quadratic polynomial) case:
- **Unique Kronecker Coefficient Matrix $\mathbf{A}_{2u} \in \mathbb{R}^{n \times \frac{n(n+1)}{2}}$**: Represents the mapping of the unique Kronecker product back to the original vector $\mathbf{x}\in\mathbb{R}^n$.
- **Kronecker Coefficient Matrix $\mathbf{A}_2 \in \mathbb{R}^{n \times n^2}$**: Represents the mapping of the standard Kronecker product back to the original vector $\mathbf{x}\in\mathbb{R}^n$.
These matrices are useful for applications in polynomial regression, symmetric tensor computations, and vectorization of symmetric matrices.
## Features
- Compute the unique Kronecker product for vectors of any dimension $n$ and any repeated (Kronecker) order $k$.
- Generate the associated polynomial and Kronecker coefficient matrices $\mathbf{A}_{ku}$ and $\mathbf{A}_k$ where $k$ is the order of the Kronecker product.
- Convert between unique and standard Kronecker products.
- Utility functions for polynomial modeling and symmetric tensor operations.
## Installation
You can install it using the command
```julia
using Pkg
Pkg.add("UniqueKronecker")
using UniqueKronecker
```
or install it directly from GitHub:
```julia
using Pkg
Pkg.add(url="https://github.com/YourUsername/UniqueKronecker.jl")
```
Replace `YourUsername` with the actual GitHub username or organization where the package is hosted.
## Usage
### Importing the Package
```julia
using UniqueKronecker
```
### Computing the Unique Kronecker Product
Compute the $k$-th order unique Kronecker product of vector `x`:
```julia
x = [2.0, 3.0, 4.0] # Example vector in ℝ³
x_unique_kron = x ⊘ x
println(x_unique_kron)
# Output: [4.0, 6.0, 8.0, 9.0, 12.0, 16.0]
# Corresponding to [x₁², x₁x₂, x₁x₃, x₂², x₂x₃, x₃²]
```
### Computing Coefficient Matrices
#### Polynomial Matrix $\mathbf{A}_2$
Compute the polynomial coefficient matrix $\mathbf{A}_2$:
```julia
n = 3
A2 = zeros(n,n^2)
for i in 1:n
x = rand(n)
A2[i,:] = kron(x,x)
end
println(A2)
# Output: A matrix of size (3, 9) for this example
```
#### Unique/Nonredundant Polynomial Coefficient Matrix $A_{2u}$
Convert the polynomial matrix $A_2$ into the unique polynomial coefficient matrix $A_{2u}$:
```julia
A2u = eliminate(A2, 2)
println(A2u)
# Output: A matrix of size (3, 6) for this example
```
This can be converted back
```julia
A2 = duplicate(A2u, 2)
println(A2)
# Output: the A2 matrix
```
To make the coefficients symmetric for redundant terms use `duplicate_symmetric`
```julia
A2s = duplicate_symmetric(A2u, 2)
println(A2s)
# Output: the A2 matrix with symmetric coefficients
```
### Relationship Between Matrices
The following relationship holds:
$$
\mathbf{A}_{2u} \cdot (\mathbf{x} \oslash \mathbf{x}) = \mathbf{A}_2 \cdot (\mathbf{x} \otimes \mathbf{x})
$$
This allows mapping between the unique Kronecker product space and the standard Kronecker product space.
### Generalizing to Higher-Order Products
Compute higher-order unique Kronecker products by specifying a higher value of $k$:
```julia
k = 3 # Third-order product
x_unique_kron_k3 = unique_kronecker(x, k) # or ⊘(x,k)
println(x_unique_kron_k3)
# Output: Corresponding unique products of order 3
```
## Applications
- **Polynomial Regression**: Efficient computation of polynomial features without redundant terms.
- **Symmetric Tensor Computations**: Simplifies operations involving symmetric tensors.
- **Model Reduction**: Construction of reduced-order models with polynomial structures.
- **Machine Learning**: Feature engineering for higher-order interactions.
## Contributing
Contributions are welcome! If you find a bug or have a feature request, please open an issue. If you'd like to contribute code, feel free to submit a pull request.
## License
This project is licensed under the [MIT License](https://github.com/smallpondtom/UniqueKronecker.jl/blob/main/LICENSE).